Find a matrix P such that P'AP orthogonally diagonalizes A. Verify that PTAP_gives the proper diagonal form. (Enter each matrix in the form [[row 1], [row 2], ...], where each row is a comma-separated list.)4 2 0 02 4 0 0A =0 0 4 20 0 2 4[1,0, – 1,0],[1, 0, 1, 0],[0, 1, 0, – 1],[0, 1, 0, 1]], [[12, 0, 0, 0],[0, 12, 0, 0],[0, 0, 4, 0],[0, 0, 0 ×)(Р, РТАР)

The given matrix is A=4200240000420024.A is a block diagonal matrix, where the block A1=A2=4224. So, A = A100A2.Then, eigenvalues of A can be determined using A-λI=A1-λIA2-λI, where λ is the eigenvalue of A.Let λ be the eigenvalue of A1. Then,A-λI=4-λ224-λ =4-λ2-4 = 16 + λ2 - 8λ - 4 = λ2 - 8λ + 12 =(λ2-6λ -2λ+12) = λ(λ-6)-2(λ-6) = (λ-6)(λ-2) So, the eigenvalues of A are 6, 6, 2, 2.Next, we find the eigenvectors corresponding to each eigenvalue.For, λ=6,4-λ20024-λ00004-λ20024-λ=-22002-20000-22002-2 =1-1000000001-10000 =1-100001-100000000Now, we solve, the equation 1-100001-100000000v1v2v3v4=0000.We get v1-v2=0; v3-v4=0. Let v1=t, v3=s. Therefore, v=ttss=1100t + 0011s.Then, the eigenvectors are 1100,0011.For λ=2, 4-λ20024-λ00004-λ20024-λ=2200220000220022 =1100001100000000 We get v1+v2=0; v3+v4=0. Let v1=t, v3=s. Therefore, v=ttss=1-100t + 001-1s.Then, the eigenvectors are 1-100,001-1.Now, we define P as P=v1 v2 v3 v4 = 10-101010010-10101 and D =λ10000λ20000λ30000λ4= 6000060000200002.Thus, we can define the diagonalization as S-1AS=D.Now, we calculate PTAP PTAP=11000011-110000-11420024000042002410-101010010-10101 =66000066-220000-2210-101010010-10101 =120000120000400004Thus, PTAP is a diagonal matrix.